Such a great word. I expect most of you are already familiar with the term, but because every time I think that someone proves me wrong, and because I think it's fantastic, I elaborate.
Mansplaining isn't just the act of explaining while male, of course; many men manage to explain things every day without in the least insulting their listeners.
Mansplaining is when a dude tells you, a woman, how to do something you already know how to do, or how you are wrong about something you are actually right about, or miscellaneous and inaccurate "facts" about something you know a hell of a lot more about than he does.
Bonus points if he is explaining how you are wrong about something being sexist!
Think about the men you know. Do any of them display that delightful mixture of privilege and ignorance that leads to condescending, inaccurate explanations, delivered with the rock-solid conviction of rightness and that slimy certainty that of course he is right, because he is the man in this conversation?
That dude is a mansplainer.
Sadly, many of these dudes are our bosses or supervisors or other authority figures to whom we cannot give much crap. But if it's someone you know in a social setting, etiquette experts agree that the appropriate thing to do is to roll your eyes and say, "Oh, please, mansplain to me some more."
You are doing him a favour. Friends don't let friends foster mansplaining.
ETA: Follow-up talking about why I don't think a gender-neutral term accurately describes the privilege behind this behaviour.
No one seems to know who came up with this term - I myself saw it in several places prior to writing this post - but the most likely antecedent is this article by Rebecca Solnit.